4 - Buildings

Summary

In the chapter we introduce (spherical) buildings. We develop the theory in some detail, sometimes providing proofs. We introduce the shadow geometries and discuss some properties of particular instances in detail. To that end we use “chain calculus”, which provides an efficient way to determine the diameter of a given shadow geometry, or the maximal distance between two generic objects of distinct type. We hence deduce that the shadow geometry of type E(6,1) yields a strongly regular graph. We provide an explicit construction of that geometry using a split octonion algebra. We also discuss the Klein correspondence, and we discuss triality, again with the aid of a split octonion algebra, and use this to construct the split Cayley hexagon over any field.We deduce a rank 4 representation of a corresponding strongly regular graph.